Faster Implied Volatilities via the Implicit Function Theorem. Michael A. Kelly* Lafayette College

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1 Faster Implie Volatilities via the Implicit Fctio Theorem Michael A. Kelly* Lafayette ollege Keywors: optios, implie volatility JEL lassificatio:, 6, G This is the pre-peer reviewe versio of the followig article: Faster Implie Volatilities via the Implicit Fctio Theorem by Michael A. Kelly. Fiacial Review * orrespoig athor: Simo eter, Room 9, Easto, PA 8-776; Phoe: 6--; kellyma@lafayette.e. I thak Satosh Nabar of JP Morga hase for helpfl commets.

2 Faster Implie Volatilities via the Implicit Fctio Theorem Abstract We preset a faster, more accrate techiqe for estimatig implie volatility sig the staar partial erivatives of the Black-Scholes optios-pricig formla. Besie Newto-Raphso a slower approximatio methos, this techiqe is the first to provie a error tolerace, which is essetial for practical applicatio. All existig oiterative approximatio methos o ot provie error toleraces a have the potetial for large errors.

3 . Itroctio Sice Black a Scholes 97 veile their optio pricig formla, practitioers a theoreticias have bee estimatig the volatility implie by optio prices. The impact has bee so profo that ivestmet professioals a acaemics ofte igore optio prices, focsig istea po the implie volatility. Whatever the impact of implie volatility, it mst be calclate. The Black- Scholes optios-pricig formla caot be iverte, so a approximatio mst be se. We preset a faster, more accrate techiqe for estimatig implie volatility sig the staar partial erivatives of the Black-Scholes optios-pricig formla. This is the first o-iterative techiqe to provie a error tolerace, which is critical sice all existig o-iterative approximatio methos have the potetial for large errors. Existig approximatio techiqes ca be classifie as follows. First, merically iterative techiqes, sch as Newto-Raphso, are comptatioally itesive, yet provie a error tolerace a ca be mae to coverge. Seco, qasi-iterative techiqes, sch as those propose by hace 996 a hambers a Nawalkha se a secoorer Taylor expasio of the call optio fctio i volatility a strike to create a qaratic i implie volatility that is solve sig the qaratic formla. hambers a Nawalkha restrict hace s Taylor expasio to be oly i volatility, improvig its accracy. These methos are qasi-iterative sice they reqire a a priori estimate of the implie volatility to serve as a startig poit. Neither metho provies a error tolerace. Thir, Breer a Sbrahmayam 988, orrao a Miller 996, a Hallerbach preset o-iterative, close form approximatios. Breer a Sbrahmayam preset a formla for calclatig the implie volatility of at-the-moey forwar optios sig a liear approximatio of the cmlative ormal istribtio fctio. orrao a Miller se a qaratic approximatio of the cmlative ormal istribtio fctio to create a seco-orer polyomial i the implie volatility that they solve sig the qaratic formla. There is o at-the-moey restrictio. Hallerbach solves a qaratic a ses approximatio techiqes to erive a more precise implie volatility estimate tha that of orrao-miller. Noe of the o-iterative methos provie a error tolerace. We take a ifferet approach i this paper. Rather tha approximatig the optio price fctio: f For istace, see ha, he, a Lg where implie volatility is relate to eqity retrs. I wol like to thak a aoymos referee for the sggeste classificatio of methos. See Jackso a Stato for the Newto-Raphso metho. See Beiga for the simpler bisectio metho. Maaster a Koehler 98 propose a start vale that they state garatees covergece sig the Newto-Raphso metho. Hallerbach poits ot that this fails for at-the-moey forwar optios. We focs o Eropea call optios, bt the aalysis ca be extee to Eropea pt optios.

4 We cosier as a fctio of. Hece, f Usig the implicit fctio theorem, we calclate the partial erivatives of to Taylor expa. higherorerterms Or procere is qasi-iterative, i the spirit of hace 996 a hambers- Nawalkha. Similar to hambers-nawalkha, we restrict or Taylor expasio to be i, for greater accracy. Importatly, a criterio exists that allows the estimate to be se oly whe a pre-specifie tolerace is met. Sice the error ca be qite large, a error tolerace is essetial for ay estimate to have practical applicatio. No previos, o-iterative metho provies a error tolerace level.. Metho We cosier the Black-Scholes optios-pricig formla for a Eropea call optio withot ivies: where rt SN Ke N l S / K r t t t with S = stock price K = strike price r = cotiosly compoe iterest rate t = time to optio matrity teor N = cmlative ormal istribtio fctio To employ the implicit fctio theorem see Hbbar a Hbbar, chap., the formla is rewritte as: rt, SN Ke N The metho ca be extee to Eropea pt optios. For Eropea call optios, the iclsio of ivies is straightforwar. The aalysis ca also be extee to icle Eropea optios o ftres a foreig exchage. For America optios, the implicit fctio techiqe col be applie to the formla propose by Roll 997, Geske 979, 98, a Whaley 98. Hll 997, p. 9 offers a smmary.

5 A Taylor expasio i two variables, S has poor covergece for ot-of-themoey optios, so we restrict orselves to a expasio i. > implies that vega. Hece, a implicit fctio,, exists for some regio aro ay positive. The erivatives of are calclate with respect to by sig the chai rle. These erivatives allow s to Taylor expa to ay orer. higherorerterms Stock a optio prices typically move simltaeosly. By oly expaig aro, we appear to limit the sefless of or approximatio techiqe. Let s cosier a applicatio to see how the techiqe col be se. Most real-time volatility calclators are market moitors. That is, each time the stock price or the optio price chages, the implie volatility is pate. Except for the first calclatio, the previos implie volatility is available to calclate the ext implie volatility. Sppose that the stock price moves from S to S. We se the ew stock price, S, a the previosly calclate implie volatility,, to calclate a Black-Scholes optio vale, S,, a its erivatives. We the calclate the partial erivatives of evalate at S, S, a calclate or estimate for volatility sig the Taylor expasio aro S, extrapolate to the observe price of the optio,. 6 Sice we ee a startig implie volatility for or calclatios, or techiqe is qasi-iterative, i the spirit of hace 996 a hambers a Nawalkha.. Reslts Estimate volatilities base po a fifth-orer expasio of are presete i Table. We have fo that a fifth-orer expasio provies, i most cases, fairly accrate estimates for a broa rage of moeyess a matrities. The optio is a - moth, at-the-moey call with implie volatility eqal to % a iterest rates eqal to zero. 7 The calclate implie volatilities i the table se % as the see. The colm heaigs represet the tre implie volatility, while the rows represet the stock price. The erivatives appear i Appeix to the fifth orer. The evalatio of the erivatives for a give S,, r, t oly reqires the staar call optio erivatives calclate by most software packages. 6 Similarly, the crret iterest rate, r, is se to calclate the Taylor expasio. 7 We se r=, so that the at-the-moey forwar is eqal to the crret spot price. The estimatio techiqe is vali for ay iterest rate.

6 Table Reslts for th orer expasio of. K = ; r = ; t =.; = %.%.% 6.% 8.%.%.%.% 7.%.% 7.%.8% 6.% 8.%.%.%.8% 7.8% % 7.6%.% 6.% 8.%.%.%.% 6.8% 9.% 8 9.%.8% 6.% 8.%.%.%.% 7.79% 6.6% 8 7.8%.% 6.% 8.%.%.%.% 7.7%.68% 9.6%.% 6.% 8.%.%.%.% 7.%.% 9.8%.% 6.% 8.%.%.%.% 7.%.%.%.% 6.% 8.%.%.%.% 7.%.%.7%.% 6.% 8.%.%.%.% 7.%.%.8%.% 6.% 8.%.%.%.% 7.%.% 6.6%.% 6.% 8.%.%.%.% 7.% 7.7% 7.89%.7% 6.% 8.%.%.%.% 7.6% 6.7% 9.%.8% 6.% 8.%.%.%.% 7.79% 6.6% 9.%.7% 6.% 8.%.%.%.% 7.79% 6.6%.88%.97% 6.9% 8.%.%.%.96% 9.9% 8.% For stock prices betwee 7 a, the estimate implie volatilities are withi.% for the rage of 8 to % tre implie volatilities. For most real-time calclatios, this is well withi the rage of implie volatility for a sigle ay. For stock prices betwee 8 a, the rage extes to to 7%. The optio premim whe the stock price is 7 is., a level low eogh to expect that the th orer approximatio wol break ow. Whe the stock price is, the optio premim over itrisic is.7, a level at which we wol expect problems with the approximatio. While the reslts of the th orer expasio are accrate over a large rage, they o evetally become highly iaccrate. For the case where S = a = %, the estimate volatility is,8%! To avoi these errors, a ititive criterio is available to etermie whether the extrapolatio is withi a particlar tolerace. Sppose we kow. The stock price moves to S a we calclate, S. For a call optio price,, the th orer estimate is withi.% tolerace if: abs abs abs,, max e rt S S S K,. 6 whe < % a t < years or,, max e rt S S S K, whe < % a t < years or,, max e rt S S S K, whe < % a t < years 8.. For a iitial volatility less tha % a a teor less tha years, if the absolte vale of the percetage chage of the call premim is less tha 6%, the extrapolatio is withi.% of the tre volatility. The call premim i the eomiator is ajste by 8 Or approximatio techiqe ca be se for > % by calclatig the appropriate criterio ct-off levels. While the techiqe may be se for t > years at a lower ct-off level, we restrict orselves to t < years sice most staar optios i the Uite States have teors less tha years.

7 sbtractig the itrisic vale of the optio ajste for the time vale of the strike i.e., the iscote itrisic or maxs e -rt K,. t-off levels ca be compte for other toleraces e.g.,.% a for higher or lower orer polyomial expasios. 9 By comptig ct-off levels for the st throgh th orer polyomials, a implie volatility calclator ee oly se the smallest egree polyomial reqire to remai withi tolerace. If the ct-off level is violate, the implie volatility calclator reverts to Newto- Raphso or some other metho. The implie volatility obtaie from Newto-Raphso the is se as the startig poit for sbseqet extrapolatios of. These ct-off levels were etermie by examiig all stock prices a volatilities where the optio premim is. above iscote itrisic. The strike se is, so the excle optios have premims less tha.% of the strike. The stock price icremet is. The volatility icremet is.% for <= %,.% for <= 7%, a % for > 7%. Varyig iterest rates has o effect o the reslts. A more etaile look at the ct-off levels appears i Figre Figre % % % % % % 7 ays 9 moths years years t-off levels for.% tolerace vs. volatility for varios optio teors The ct-off levels show i Figre appear to have a well-behave patter that might be etermie by sig some reslts of arr. arr has calclate the geeral soltios for all th orer erivatives of the Black-Scholes optios-pricig formla with respect to stock price, volatility, iterest rate, time, a ivie yiel. A fctioal form for the error term might be fo by sig arr s th orer erivative eqatios to estimate the error term of the Taylor expasio. 9 For a th orer approximatio, the ct-off level for.% tolerace is. for < % a t < years,. for < % a t < years, a.7 for < % a t < years. For a st orer approximatio liear, the ct-off level for a.% tolerace is. for < % a t < years. 6

8 arr also shows that the th orer Taylor expasio i volatility coverges as icreases oly i a rais aro. We speclate that a similar rais exists aro for the Taylor expasio of the implicit fctio,, a that the tolerace levels are strictly withi this rais. The existece of a fiite covergece rais is likely why a global approximatio for implie volatility has ot bee fo.. ompariso to previos reslts The th orer approximatio of shol compare well to o-iterative approximatios, sch as those of orrao-miller 996 a Hallerbach, for three reasos. For the th orer approximatio, the stock price is fixe while the o-iterative approximatios are global approximatios that are goo for ay stock price a o ot extrapolate from a iitial volatility. Also, the th orer approximatio has a error tolerace, assrig agaist large errors i the estimate. We have compare the th orer approximatio to the Hallerbach approximatio a fi that the th orer approximatio is cosierably more accrate for a wie rage of matrities a moeyess. For applicatios for which a qasi-iterative approximatio is appropriate, the th orer approximatio is preferre to the orrao- Miller 996 a Hallerbach approximatios. However, Hallerbach s estimates are qite accrate for a wie rage of volatilities, moeyess, a matrities. His estimate is a goo potetial see for or qasi-iterative techiqe. We ow compare the th orer estimate to that of hace 996, a hambers a Nawalkha. hambers a Nawalkha covert hace s two-variable estimator ito a qasi-iterative, ivariate estimator, like the th orer approximatio. No error tolerace for their estimate exists. Table shows the compariso of the hace- hambers-nawalkha estimate to the th orer estimate. Table illstrates that the th orer approximatio is highly accrate for optios that are ear-the-moey forwar, hece we focs o optios that are i a ot-of-the-moey forwar. ase is a ot-ofthe-moey, extremely short-ate optio. ase is a eep ot-of-the-moey, meim matrity optio. ase is a i-the-moey forwar, log-ate optio. For a oe-week optio, a % ifferece betwee the strike a stock price is fairly ot-of-the-moey. While Hallerbach lists his reslts solely base po the aggregate volatility, t, we caot sice we have take erivatives with respect to. 7

9 Table ompariso to hace-hambers-nawalkha. K = ; = %; r = % Tre Vol Stock Price = 98 Stock Price = 8 Stock Price = Teor =.9 Oe week Teor =. Teor = hace- hace- hace- hambers- riterio hambers- riterio hambers- Nawalkha th Orer Vale Nawalkha th Orer Vale Nawalkha th Orer riterio Vale %.79%.88%.77 7.% 6.68%.96.6%.%.679 %.7%.%..8%.7%.7.8%.%.67 %.7%.%.79.6%.%.78.8%.%. %.%.%..%.%..%.%. %.988%.% %.%.88.99%.%.9 % 9.9%.9% %.%.7 9.9%.%.67 %.768%.9% % 7.7%.76.8%.7%.6886 As expecte the th orer approximatio otperforms the hambers-nawalkha approximatio; however, for may volatility vales, the ifferece is slight. The th orer approximatio shows the most improvemet whe extrapolatig owwar.. oclsio The implicit fctio theorem a Taylor s theorem are powerfl tools for estimatig the implie volatility fctio. The estimatio of is accrate for a wie rage of teors, volatilities, a iterest rates. A criterio exists so that the estimator is se oly whe a tolerace is satisfie. The th orer Taylor expasio of shol be of particlar se for real-time implie volatility calclators. If greater precisio is eee, a higher orer Taylor expasio ca be calclate. The simple criterio of the call optio premim movig by o more tha a fixe percetage of iscote itrisic is itrigig a poits to the potetial of sig this techiqe to gai a greater isight ito the behavior of the implie volatility fctio. 8

10 9 Appeix We preset the partial erivatives of. To calclate the first partial erivative with respect to, we employ the chai rle o, : Sice a = vega, vega The higher-orer erivatives follow. The cross-partials of with respect to a as well as the seco-orer a higher partials with respect to are zero: / / 6 / / where vega vega vega / / vega / vega Note that all partials of with respect to are a fctio of,,, t, a vega. Hece, the existig ata moel of most optio software packages ee ot be chage to accommoate these calclatios. E Appeix Note that a or, alteratively, t a!! t for >.

11 Refereces Beiga, Simo,. Fiacial Moelig, Eitio. MIT Press, ambrige, Massachsetts. Black, Fischer a Myro Scholes, 97. The pricig of optios a corporate liabilities, Joral of Political Ecoomy 8, Breer, Meachem a Marti Sbrahmayam, 988. A simple formla to compte the implie staar eviatio, Fiacial Aalyst Joral, 8-8. arr, Peter,. Derivig erivatives of erivative secrities, Joral of omptatioal Fiace, -9. hambers, Doal a Sajay Nawalkha,. A improve approach to comptig implie volatility, The Fiacial Review 6, 89-. ha, Kam, Lois heg a Peter Lg,. Asymmetric volatility a traig activity i iex ftres optios, The Fiacial Review, 8-7. hace, Do, 996. A geeralize simple formla to compte the implie volatility, The Fiacial Review, orrao, harles a Thomas Miller, 996. A ote o a simple, accrate formla to compte implie staar eviatios, Joral of Bakig a Fiace, 9-6. Geske, Robert, 98. ommets o Whaley s ote, Joral of Fiacial Ecoomics 9, -. Geske, Robert, 979. A ote o a aalytical valatio formla for protecte America call optios o stocks with kow ivies, Joral of Fiacial Ecoomics 7, 7-8. Hallerbach, Wifrie,. A improve estimator for Black-Scholes-Merto implie volatility, Workig Paper, Erasms Uiversiteit Rotteram,. Hbbar, Joh a Barbara Hbbar,. Vector alcls, Lier Algebra, a Differetial Forms, Eitio. Pretice Hall, New York, New York. Hll, Joh, 997. Optios, Ftres, a Other Derivatives, r Eitio. Pretice Hall, New York, New York. Jackso, Mary a Mike Stato,. Avace Moellig i Fiace sig Excel a VBA, Joh Wiley & Sos, West Sssex, Egla.

12 Roll, Richar, 977. A aalytical formla for protecte America call optios o stocks with kow ivies, Joral of Fiacial Ecoomics, -8. Whaley, Robert, 98. O the valatio of America call optios o stocks with kow ivies, Joral of Fiacial Ecoomics 9, 7-.

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